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Baigent, Stephen, Hou, Zhanyuan, Elaydi, Saber, Balreira, E. C. and Luís, Rafael (2023) A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds. Journal of Difference Equations with Applications, 29 (5). pp. 575-591. ISSN 1023-6198 (print) 1563-5120 (online)
A quadratic Lyapunov function is demonstrated for the noninvertible planar Ricker map (x, y) → (xe^{r−x−αy}, ye^{s−y−βx}) which shows that for α,β > 0, and 0 < r, s ≤ 2 all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of (0,∞)^2). Our approach bypasses and improves on methods that rely on monotonicity, which require 0 < r, s ≤ 1. We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle.
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